An Interval Constraint Programming Approach for Quasi Capture Tube Validation

Authors Abderahmane Bedouhene, Bertrand Neveu, Gilles Trombettoni, Luc Jaulin, Stéphane Le Menec

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Abderahmane Bedouhene
  • LIGM, Ecole des Ponts ParisTech, Université Gustave Eiffel, CNRS, Marne-la-Vallée, France
Bertrand Neveu
  • LIGM, Ecole des Ponts ParisTech, Université Gustave Eiffel, CNRS, Marne-la-Vallée, France
Gilles Trombettoni
  • LIRMM, Université de Montpellier, CNRS, France
Luc Jaulin
  • Lab-STICC, ENSTA-Bretagne, Brest, France
Stéphane Le Menec
  • MBDA, Le Plessis Robinson, France


We also thank our colleagues, Alexandre Goldsztejn and Alessandro Colotti, for the exchange of ideas and their kind help on the experiments.

Cite AsGet BibTex

Abderahmane Bedouhene, Bertrand Neveu, Gilles Trombettoni, Luc Jaulin, and Stéphane Le Menec. An Interval Constraint Programming Approach for Quasi Capture Tube Validation. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Proving that the state of a controlled nonlinear system always stays inside a time moving bubble (or capture tube) amounts to proving the inconsistency of a set of nonlinear inequalities in the time-state space. In practice however, even with a good intuition, it is difficult for a human to find such a capture tube except for simple examples. In 2014, Jaulin et al. established properties that support a new interval approach for validating a quasi capture tube, i.e. a candidate tube (with a simple form) from which the mobile system can escape, but into which it enters again before a given time. A quasi capture tube is easy to find in practice for a controlled system. Merging the trajectories originated from the candidate tube yields the smallest capture tube enclosing it. This paper proposes an interval constraint programming solver dedicated to the quasi capture tube validation. The problem is viewed as a differential CSP where the functional variables correspond to the state variables of the system and the constraints define system trajectories that escape from the candidate tube "for ever". The solver performs a branch and contract procedure for computing the trajectories that escape from the candidate tube. If no solution is found, the quasi capture tube is validated and, as a side effect, a corrected smallest capture tube enclosing the quasi one is computed. The approach is experimentally validated on several examples having 2 to 5 degrees of freedom.

Subject Classification

ACM Subject Classification
  • Applied computing → Operations research
  • Mathematics of computing → Ordinary differential equations
  • Mathematics of computing → Differential algebraic equations
  • Mathematics of computing → Interval arithmetic
  • Theory of computation → Constraint and logic programming
  • Constraint satisfaction problem
  • Interval analysis
  • Dynamical systems
  • Contractor


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